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Градиентный бустинг. Реализация с нуля на Python и разбор особенностей его модификаций (XGBoost, CatBoost, LightGBM)

На сегодняшний день градиентный бустинг (gradient boosting machine) является одним из основных production-решений при работе с табличными, неоднородными данными, поскольку обладает высокой производительностью и точностью, а если быть точнее, то его модификации, речь о которых пойдёт чуть позже.

В данной статье представлена не только реализация градиентного бустинга GBM с нуля на Python, но а также довольно подробно описаны ключевые особенности его наиболее популярных модификаций.

Ноутбук с данными алгоритмами можно загрузить на Kaggle (eng) и GitHub (rus).

Содержание

Gradient Boosting Machine

Gradient Boosting Machine

Как и в случае с Adaboost, градиентный бустинг добавляет базовые модели в ансамбль последовательно, однако вместо обучения моделей с учётом весов на основе ошибок предшественников, в данном случае модели обучаются на остаточных ошибках (residual errors), допущенных предыдущими моделями.

Принцип работы градиентного бустинга для регрессии

Алгоритм строится следующим образом:

1) первоначальному прогнозу присваивается среднее значение y_train для всех образцов;
2) рассчитываются остатки модели на основе антиградиента функции потерь;
3) регрессионное дерево обучается на X_train и остатках, далее делается прогноз на X_train;
4) полученный прогноз добавляется к первоначальному и шаги 2-4 повторяются для каждого дерева;
5) после обучения всех моделей снова создаётся первоначальный прогноз из шага 1;
6) далее делаются прогнозы для X_test на обученных деревьях и добавляются к первоначальному;
7) полученная сумма и будет конечным прогнозом.

Формулы для расчётов

Функция потерь: $L(y_i, F(x_i)) = \frac{1}{2} (y_i - F(x_i))^2$

Остатки: $r_{ik} = -\left[\frac{\partial L(y_i, F(x_i))}{\partial F(x_i)}\right]_ {F(x) = F_{m-1}(x)} = y_{i} - F_{m-1}(x_i)$

Принцип работы градиентного бустинга для классификации

В случае классификации дела обстоят немного сложнее: к каждому уникальному классу необходимо применить one-hot encoding и перевести в вероятности с помощью softmax-функции, а к прогнозам деревьев добавить коэффициент $\gamma$ , который регулирует степень вклада каждого нового дерева в общую модель для снижения переобучения. Также качество прогнозов градиентного бустинга можно существенно улучшить, применив концепцию K-class LogitBoost: для каждого дерева рассчитываются веса, а только потом на их основе остатки.

Алгоритм строится следующим образом:

1) для y_train применяется one-hot encoding и первоначальному прогнозу приcваиваются значения 0 для каждого класса;
2) прогнозы преобразуются в вероятности с помощью softmax-функции;
3) рассчитываются остатки модели на основе антиградиента функции потерь и вероятностей (а также веса в случае LogitBoost);
4) регрессионное дерево обучается на X_train и остатках (а в случае LogitBoost ещё присваиваются веса), далее делается прогноз на X_train;
5) для каждого листа в дереве рассчитываются коэффициенты $\gamma$ на основе остатков, взятых по позициям наблюдений, попавших в определённый в листовой узел;
6) полученные прогнозы для каждого класса и сумма коэффициентов $\gamma$ добавляются к первоначальным;
7) шаги 2-6 повторяются для каждого дерева в каждом классе;
8) после обучения всех моделей создаётся первоначальный прогноз из шага 1;
9) далее делаются прогнозы для X_test на обученных деревьях по каждому классу и добавляются к первоначальным;
10) классы с максимальной суммой и будут конечным прогнозом.

Формулы для расчётов

Функция потерь:

$L(\{y_k, F_k(x)\}_{1}^{K}) = -\sum\limits_{k=1}^{K} y_k \ log \ p_k(x)$

Вероятность каждого класса:

$p_k(x) = softmax(F_k(x)) = \frac{e^{F_k(x)}}{\sum\limits_{l=1}^{K} e^{F_l(x)}}$

Остатки:

$r_{ik} = -\left[\frac{\partial L(\{y_{il}, F_l(x_i)\}_{l=1}^{K})}{\partial F_k(x_i)}\right]_ {\{F_l(x) = F_{l,m-1}(x)\}_{1}^{K}} = y_{ik} - p_{k,m-1}(x_i)$

Коэффициент $\gamma$ :

$\gamma_{jkm} = \frac{K-1}{K} \frac{\sum_{x_i \in R_{jkm}} r_{ik}} {\sum_{x_i \in R_{jkm}} |r_{ik}| (1 - |r_{ik}|)}$

Веса для K-class LogitBoost:

Остатки для K-class LogitBoost:

$r_{ik} = \frac{K-1}{K} \frac{y_{ik} - p_k(x_i)}{w_k(x_i)}$

Импорт необходимых библиотек

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import load_diabetes
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score, mean_absolute_percentage_error
from sklearn.tree import DecisionTreeRegressor
from sklearn.ensemble import GradientBoostingClassifier, GradientBoostingRegressor
from sklearn.ensemble import HistGradientBoostingClassifier, HistGradientBoostingRegressor
from mlxtend.plotting import plot_decision_regions

Реализация на Python с нуля

class GBMClassifier:
    def __init__(self, logitboost=False, learning_rate=0.1, n_estimators=100,
                 max_depth=3, random_state=0):
        self.logitboost = logitboost
        self.learning_rate = learning_rate
        self.n_estimators = n_estimators
        self.max_depth = max_depth
        self.random_state = random_state

    def _softmax(self, predictions):
        exp = np.exp(predictions)

        return exp / np.sum(exp, axis=1, keepdims=True)

    def _compute_gammas(self, residuals, leaf_indexes, eps=1e-4):
        gammas = []

        for j in np.unique(leaf_indexes):
            x_i = np.where(leaf_indexes == j)
            numerator = np.sum(residuals[x_i])
            norm_residuals_xi = np.linalg.norm(residuals[x_i]) + eps
            denominator = np.sum(norm_residuals_xi * (1 - norm_residuals_xi))
            gamma = (self.K - 1) / self.K * numerator / denominator
            gammas.append(gamma)

        return gammas

    def fit(self, X, y):
        self.K = len(np.unique(y))
        self.trees = {k: [] for k in range(self.K)}
        one_hot_y = pd.get_dummies(y).to_numpy()   # one-hot encoding
        predictions = np.zeros(one_hot_y.shape)

        for _ in range(self.n_estimators):
            probabilities = self._softmax(predictions)

            for k in range(self.K):
                if self.logitboost:   # based on K-class LogitBoost
                    numerator = (one_hot_y.T[k] - probabilities.T[k])
                    denominator = probabilities.T[k] * (1 - probabilities.T[k])
                    residuals = (self.K - 1) / self.K * numerator / denominator
                    weights = denominator
                else:
                    residuals = one_hot_y.T[k] - probabilities.T[k]
                    weights = None

                tree = DecisionTreeRegressor(criterion='friedman_mse', max_depth=self.max_depth,
                                             random_state=self.random_state)
                tree.fit(X, residuals, sample_weight=weights)
                self.trees[k].append(tree)

                leaf_indexes = tree.apply(X)
                gammas = [] if self.logitboost else self._compute_gammas(residuals, leaf_indexes)
                predictions.T[k] += self.learning_rate * tree.predict(X) + np.sum(gammas)

    def predict(self, samples):
        predictions = np.zeros((len(samples), self.K))

        for i in range(self.n_estimators):
            for k in range(self.K):
                predictions.T[k] += self.learning_rate * self.trees[k][i].predict(samples)

        return np.argmax(predictions, axis=1)

class GBMRegressor:
    def __init__(self, learning_rate=0.1, n_estimators=100, max_depth=3, random_state=0):
        self.learning_rate = learning_rate
        self.n_estimators = n_estimators
        self.max_depth = max_depth
        self.random_state = random_state
        self.trees = []

    def fit(self, X, y):
        self.initial_leaf = y.mean()
        predictions = np.zeros(len(y)) + self.initial_leaf

        for _ in range(self.n_estimators):
            residuals = y - predictions
            tree = DecisionTreeRegressor(criterion='friedman_mse', max_depth=self.max_depth,
                                         random_state=self.random_state)
            tree.fit(X, residuals)
            predictions += self.learning_rate * tree.predict(X)
            self.trees.append(tree)

    def predict(self, samples):
        predictions = np.zeros(len(samples)) + self.initial_leaf

        for i in range(self.n_estimators):
            predictions += self.learning_rate * self.trees[i].predict(samples)

        return predictions

Код для отрисовки решающих границ

def decision_boundary_plot(X, y, X_train, y_train, clf, feature_indexes, title=None):
    feature1_name, feature2_name = X.columns[feature_indexes]
    X_feature_columns = X.values[:, feature_indexes]
    X_train_feature_columns = X_train.values[:, feature_indexes]
    clf.fit(X_train_feature_columns, y_train.values)

    plot_decision_regions(X=X_feature_columns, y=y.values, clf=clf)
    plt.xlabel(feature1_name)
    plt.ylabel(feature2_name)
    plt.title(title)

Загрузка датасетов

Для обучения моделей будет использован Glass Classification датасет, где необходимо верно определить тип стекла по его признакам. В случае регрессии используется Diabetes датасет из scikit-learn.

Данные ниже будут использоваться в примерах для всех модификаций градиентного бустинга.

df_path = "/content/drive/MyDrive/glass.csv"
glass_df = pd.read_csv(df_path)
X1, y1 = glass_df.iloc[:, :-1], glass_df.iloc[:, -1]
y1 = pd.Series(LabelEncoder().fit_transform(y1))
X1_train, X1_test, y1_train, y1_test = train_test_split(X1, y1, random_state=0)
print(glass_df)


          RI     Na    Mg    Al     Si     K    Ca    Ba   Fe  Type
0    1.52101  13.64  4.49  1.10  71.78  0.06  8.75  0.00  0.0     1
1    1.51761  13.89  3.60  1.36  72.73  0.48  7.83  0.00  0.0     1
2    1.51618  13.53  3.55  1.54  72.99  0.39  7.78  0.00  0.0     1
3    1.51766  13.21  3.69  1.29  72.61  0.57  8.22  0.00  0.0     1
4    1.51742  13.27  3.62  1.24  73.08  0.55  8.07  0.00  0.0     1
..       ...    ...   ...   ...    ...   ...   ...   ...  ...   ...
209  1.51623  14.14  0.00  2.88  72.61  0.08  9.18  1.06  0.0     7
210  1.51685  14.92  0.00  1.99  73.06  0.00  8.40  1.59  0.0     7
211  1.52065  14.36  0.00  2.02  73.42  0.00  8.44  1.64  0.0     7
212  1.51651  14.38  0.00  1.94  73.61  0.00  8.48  1.57  0.0     7
213  1.51711  14.23  0.00  2.08  73.36  0.00  8.62  1.67  0.0     7

[214 rows x 10 columns]

X2, y2 = load_diabetes(return_X_y=True, as_frame=True)
X2_train, X2_test, y2_train, y2_test = train_test_split(X2, y2, random_state=0)
print(X2, y2, sep='\n')


          age       sex       bmi        bp        s1        s2        s3  \
0    0.038076  0.050680  0.061696  0.021872 -0.044223 -0.034821 -0.043401   
1   -0.001882 -0.044642 -0.051474 -0.026328 -0.008449 -0.019163  0.074412   
2    0.085299  0.050680  0.044451 -0.005670 -0.045599 -0.034194 -0.032356   
3   -0.089063 -0.044642 -0.011595 -0.036656  0.012191  0.024991 -0.036038   
4    0.005383 -0.044642 -0.036385  0.021872  0.003935  0.015596  0.008142   
..        ...       ...       ...       ...       ...       ...       ...   
437  0.041708  0.050680  0.019662  0.059744 -0.005697 -0.002566 -0.028674   
438 -0.005515  0.050680 -0.015906 -0.067642  0.049341  0.079165 -0.028674   
439  0.041708  0.050680 -0.015906  0.017293 -0.037344 -0.013840 -0.024993   
440 -0.045472 -0.044642  0.039062  0.001215  0.016318  0.015283 -0.028674   
441 -0.045472 -0.044642 -0.073030 -0.081413  0.083740  0.027809  0.173816   

           s4        s5        s6  
0   -0.002592  0.019907 -0.017646  
1   -0.039493 -0.068332 -0.092204  
2   -0.002592  0.002861 -0.025930  
3    0.034309  0.022688 -0.009362  
4   -0.002592 -0.031988 -0.046641  
..        ...       ...       ...  
437 -0.002592  0.031193  0.007207  
438  0.034309 -0.018114  0.044485  
439 -0.011080 -0.046883  0.015491  
440  0.026560  0.044529 -0.025930  
441 -0.039493 -0.004222  0.003064  

[442 rows x 10 columns]
0      151.0
1       75.0
2      141.0
3      206.0
4      135.0
       ...  
437    178.0
438    104.0
439    132.0
440    220.0
441     57.0
Name: target, Length: 442, dtype: float64

Обучение моделей и оценка полученных результатов

В случае классификации ручная реализация немного отличается в плане точности от scikit-learn — это связано с тем, что в последней применяется более усложнённый подход для прогнозирования и расчёта остатков, но суть остаётся прежней, и в большинстве случаев работает примерно также как и ручная реализация, особенно в сравнении с LogitBoost модификацией.

В случае регрессии, ошибка в ручной реализации даже немного ниже, чем в scikit-learn — различия незначительны.

Также стоит отметить, что в scikit-learn есть особая версия градиентного бустинга — HistGradientBoosting на основе LightGBM, предназначенная для работы с большими наборами данных, которая отрабатывает гораздо быстрее и чаще лучше, чем классический градиентный бустинг.

Полученные результаты приведены ниже.

GBMClassifier

gbc = GBMClassifier(random_state=0)
gbc.fit(X1_train, y1_train)
gbc_pred_res = gbc.predict(X1_test)
gbc_accuracy = accuracy_score(gbc_pred_res, y1_test)
print(f'gbc accuracy: {gbc_accuracy}')
print(gbc_pred_res)


gbc accuracy: 0.6111111111111112
[5 0 1 4 1 1 0 1 1 1 1 0 1 1 1 5 0 1 1 1 5 0 5 5 1 0 5 0 1 0 0 4 1 0 0 0 0
 0 0 5 1 4 1 0 0 1 0 1 0 1 0 3 5 0]

GBMClassifier с концепцией LogitBoost

logit_bc = GBMClassifier(logitboost=True, random_state=0)
logit_bc.fit(X1_train, y1_train)
logit_bc_pred_res = logit_bc.predict(X1_test)
logit_bc_accuracy = accuracy_score(logit_bc_pred_res, y1_test)
print(f'logit_gbc accuracy: {logit_bc_accuracy}')
print(logit_bc_pred_res)


logit_gbc accuracy: 0.6666666666666666
[5 0 1 4 1 1 0 1 1 1 1 0 0 1 1 5 2 1 1 0 3 0 5 1 2 0 5 0 0 1 0 0 1 0 0 0 0
 2 0 5 1 4 1 0 1 1 0 1 0 1 0 1 5 0]

GradientBoostingClassifier (scikit-learn)

sk_gbc = GradientBoostingClassifier(random_state=0)
sk_gbc.fit(X1_train, y1_train)
sk_gbc_pred_res = sk_gbc.predict(X1_test)
sk_gbc_accuracy = accuracy_score(sk_gbc_pred_res, y1_test)
print(f'sk_gbc accuracy: {sk_gbc_accuracy}')
print(sk_gbc_pred_res)

feature_indexes = [1, 3]
title1 = 'GradientBoostingClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, sk_gbc, feature_indexes, title1)


sk_gbc accuracy: 0.6666666666666666
[5 0 1 4 1 1 0 1 1 1 1 0 1 1 1 5 2 1 2 0 1 0 5 5 0 0 5 0 0 1 0 0 1 0 1 0 0
 0 0 5 1 1 1 0 1 1 0 1 0 1 0 1 5 0]

HistGradientBoostingClassifier (scikit-learn)

sk_hist_gbc = HistGradientBoostingClassifier(random_state=0)
sk_hist_gbc.fit(X1_train, y1_train)
sk_hist_gbc_pred_res = sk_hist_gbc.predict(X1_test)
sk_hist_gbc_accuracy = accuracy_score(sk_hist_gbc_pred_res, y1_test)
print(f'sk_hist_gbc accuracy: {sk_hist_gbc_accuracy}')
print(sk_hist_gbc_pred_res)

feature_indexes = [1, 3]
title2 = 'HistGradientBoostingClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, sk_hist_gbc, feature_indexes, title2)


sk_hist_gbc accuracy: 0.7407407407407407
[5 0 1 4 1 1 0 1 1 1 0 1 0 1 1 5 2 1 0 2 3 0 5 5 0 0 5 0 1 1 0 0 1 0 0 0 0
 2 0 5 1 4 1 0 0 1 0 1 0 1 0 1 5 0]

GBMRegressor

gbr = GBMRegressor(random_state=0)
gbr.fit(X2_train, y2_train)
gbr_pred_res = gbr.predict(X2_test)
mape = mean_absolute_percentage_error(gbr_pred_res, y2_test)
print(f'gbr mape: {mape}')
print(gbr_pred_res)


gbr mape: 0.3422881238725283
[255.26058697 236.87383554 189.22369747 111.4291292  195.31958776
 249.2293251   97.46956406 205.91110801 133.47952304 240.88679883
 187.19018483 171.09963906 132.63933991  86.28964353 286.56077589
 105.78347251 161.94686043  72.58069452 105.92470096 236.52718059
 178.31146418 127.93036968 164.16563703 147.68681371 229.72747212
 194.16656023 142.44300193  68.79177464 237.21638657 163.36730299
 228.0338107   85.82843049 122.42781509 153.99332065 133.44848601
 159.00754731 151.98981608 147.1238119   88.56372077 200.84785209
 111.32129613 157.60108633 137.94395538 173.39402287 176.29284757
  62.85019316 109.646804   124.14982054  83.45697583 266.40618239
 141.2307366   42.05033124 163.57898214 149.92213387 230.37326368
 226.98842404 176.95426024 128.73302071  94.7641018  180.13853236
 242.31821245 147.52903035 130.45931832  97.32617479 246.70709949
 155.26338705  88.36785447 243.39330218 215.93042351  57.89243255
  87.76236479 138.21222825 101.23449416 140.80521012 128.63575744
 180.15062966 126.14000544 220.28212417 272.05642136 203.4384734
 135.96206295 180.87259854  55.62066017 241.84218009 137.19320157
  85.57441147 154.16763601 198.15465153 107.34659467 158.72672412
 102.17554103 114.4602227   70.44016611 165.66282053 108.02991765
  87.88855231 218.47152891 250.1208086  131.93276827 164.8880371
 174.16393905 119.21512394 150.9405975   93.44225978 234.7444088
 139.46581273 248.28728947 276.8291188  121.36081625  77.12248695
 252.1021757 ]

GradientBoostingRegressor (scikit-learn)

sk_gbr = GradientBoostingRegressor(random_state=0)
sk_gbr.fit(X2_train, y2_train)
sk_gbr_pred_res = sk_gbr.predict(X2_test)
sk_mape = mean_absolute_percentage_error(sk_gbr_pred_res, y2_test)
print(f'sk_gbr mape: {sk_mape}')
print(sk_gbr_pred_res)


sk_gbr mape: 0.34315075601365186
[255.26058697 236.87383554 189.22369747 111.4291292  195.31958776
 249.2293251   97.46956406 205.91110801 133.47952304 240.88679883
 187.19018483 171.09963906 132.63933991  86.28964353 286.56077589
 105.78347251 161.94686043  72.58069452 105.92470096 236.52718059
 182.08417504 127.93036968 164.16563703 147.68681371 229.72747212
 194.16656023 142.44300193  68.79177464 237.21638657 163.36730299
 223.14955275  85.82843049 122.42781509 153.99332065 133.44848601
 159.00754731 151.98981608 147.1238119   88.56372077 200.84785209
 117.71059889 157.60108633 137.94395538 173.39402287 176.29284757
  62.85019316 109.646804   124.14982054  83.45697583 266.40618239
 141.2307366   42.05033124 163.57898214 149.92213387 230.37326368
 226.98842404 176.95426024 128.73302071  94.7641018  180.13853236
 242.31821245 147.52903035 131.71442058  97.32617479 246.70709949
 155.26338705  88.36785447 243.39330218 215.93042351  54.04270734
  87.76236479 138.21222825 101.23449416 140.80521012 128.63575744
 180.15062966 126.14000544 220.28212417 272.05642136 203.4384734
 135.96206295 180.87259854  55.62066017 241.84218009 137.19320157
  85.57441147 154.16763601 198.15465153 107.34659467 158.72672412
 102.17554103 114.4602227   70.44016611 165.66282053 108.02991765
  87.88855231 218.47152891 247.93843617 131.93276827 164.8880371
 174.16393905 119.21512394 150.9405975   93.44225978 234.7444088
 139.46581273 248.28728947 286.90898175 121.36081625  77.12248695
 252.1021757 ]

HistGradientBoostingRegressor (scikit-learn)

sk_hist_gbr = HistGradientBoostingRegressor(random_state=0)
sk_hist_gbr.fit(X2_train, y2_train)
sk_hist_gbr_pred_res = sk_hist_gbr.predict(X2_test)
sk_hist_mape = mean_absolute_percentage_error(sk_hist_gbr_pred_res, y2_test)
print(f'sk_hist_gbr mape: {sk_hist_mape}')
print(sk_hist_gbr_pred_res)


sk_hist_gbr mape: 0.3625770694795175
[264.59007545 243.59190524 180.62442773 108.53165744 195.17695513
 258.1710931  112.67894454 239.18006052 144.50027011 244.02461776
 180.19087697 171.2628973  142.6966707   94.26236013 301.44493762
  90.25622203 154.82566511  75.43291346  96.36987316 230.14565012
 196.50928346 134.41863815 167.28719774 118.51218378 209.75431111
 176.46248808 113.40188129  68.47515467 248.57638999 116.15751993
 195.54397416  59.94733366 153.03012884 165.84819847 135.95402508
 133.54279177 190.26058998 153.08963938  86.65214731 185.07312005
 110.00197459 142.99434304 111.32566034 205.63519803 163.76035901
  68.26003406 118.15074567 101.01137587  57.54356002 247.29839792
 140.32714196  41.87878548 149.89720974 189.08931404 251.48016218
 175.76119949 160.38118455 110.76716142 107.00114228 154.75409577
 256.3194536  148.8935956  119.32752592  78.65118163 252.18315433
 122.833807    92.10033912 219.56310647 199.35553751  87.27114699
  78.83529151 114.14943246  86.07433503 110.44443265 126.1152782
 193.58960355 125.06171193 207.62012728 297.41734601 206.57210109
 121.92170504 194.37322797  69.66978916 260.97214455  99.77944002
 104.34927662 131.51094535 200.0857846  114.70076625 144.92225838
 107.08233694  84.70304021  65.20782844 156.77203859  93.87016441
 117.10361355 213.79169982 258.68468943 129.45477593 140.06443861
 176.72311312 107.95798342 172.45179867 115.75603211 240.83612061
 102.2339954  249.6969811  293.80624063 110.47825419  73.86099524
 238.42985813]

Преимущества и недостатки градиентного бустинга (GBM)

Преимущества:

высокая точность;
хорошо работает с неоднородными данными;
лежит в основе других модификаций градиентного бустинга.

Недостатки:

низкая скорость обучения при работе с большими датасетами;
отсутствие возможности работы с категориальными признаками;
возможность переобучения из-за плохой предобработки данных.

Стоит отметить, что вышеперечисленные недостатки были устранены частично либо полностью в модификациях градиентного бустинга, про которые сейчас и пойдёт речь.

Дополнительные источники

Статья «Greedy Function Approximation: A Gradient Boosting Machine», Jerome H. Friedman.

Документация:

Лекции:

Пошаговое построение Gradient Boosting:

Особенности XGBoost

XGBoost (Extreme Gradient Boosting) является, пожалуй, самой популярной модификацией градиентного бустинга на сегодняшний день, с помощью которой было выиграно множество соревнований на Kaggle. Рассмотрим причины по которым так произошло.

В отличие от GBM в XGboost изначальный прогноз регрессионного дерева устанавливается f_0 (x) = 0.5 и вместо пней с глубиной 3 берутся более глубокие деревья с листьями в диапазоне от 8 до 32 как в случае регрессии, так и классификации, а к функции потерь добавляется регуляризация (L1 или L2 на выбор), которая штрафует дерево с высокой суммой нормы значений в листьях. Данная сумма представлена в виде целевой функции, расчёт значений которой аппроксимируется разложением в ряд Тейлора 2-го порядка.

Формулы для расчётов

Целевая функция:

$Obj^{\ t} = \sum\limits_{i=1}^{N} L(y_i, \hat{y_i}^{(t-1)} + f_t(x_i)) + \Omega(f_t)$

L2-регуляризация:

$\Omega(f_t) = \gamma \ T + \frac{1}{2} \lambda||w||^2$

Аппроксимация целевой функции разложением в ряд Тейлора 2-го порядка:

$Obj^{\ t} \approx \sum\limits_{i=1}^{N} [L(y_i, \hat{y_i}^{(t-1)}) + g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i) ] + \gamma \ T + \frac{1}{2} \lambda \sum\limits_{j=1}^{T} w_j^2 = \\ = \sum\limits_{j=1}^{T} [(\sum\limits_{i \in I_j}g_i) w_j + \frac{1}{2} (\sum\limits_{i \in I_j}h_i + \lambda) w_j^2 ] + \gamma \ T$

Где:

градиент первого порядка: ;
градиент второго порядка (гессиан): ;
параметры регуляризации: $\gamma, \lambda$ .

Приближение второго порядка удобно с вычислительной точки зрения, поскольку большинство членов одинаковы на каждой отдельной итерации и большая часть выражения может быть вычислена один раз и повторно использована как константа для всех остальных разбиений, и в таком случае единственное, что остаётся вычислить — это f_t(x_i) и $\Omega(f_t)$ .

Оптимальное значение веса j-го листового узла в дереве рассчитывается следующим образом:

$w^*_j = -\frac{\sum_{i \in I_j} g_i}{\sum_{i \in I_j} h_i + \lambda}$

Мера загрязнённости представляет из себя качество структуры дерева q и рассчитывается следующим образом:

$Obj^{\ t} = -\frac{1}{2} \sum\limits_{j=1}^{T} \frac{(\sum_{i \in I_j} g_i)^2}{\sum_{i \in I_j} h_i + \lambda} + \gamma \ T$

Чем меньше это значение, тем лучше структура дерева.

Поскольку перечисление всевозможных древовидных структур и выбор наилучшей весьма осложнён, мы можем оптимизировать данную задачу, добавляя по одному разбиению узла дерева за раз. В частности, мы итеративно добавляем ветви к дереву, используя жадный алгоритм, а информационный прирост будет выглядеть следующим образом:

$gain = \frac{1}{2} \left[\frac{(\sum_{i \in I_L} g_i)^2}{\sum_{i \in I_L} h_i + \lambda} + \frac{(\sum_{i \in I_R} g_i)^2}{\sum_{i \in I_R} h_i + \lambda} - \frac{(\sum_{i \in I} g_i)^2}{\sum_{i \in I} h_i + \lambda} \right] - \gamma$

Информационный прирост будет показывать изменение чистоты узла после его разбиения. Его максимальное значение будет соответствовать наилучшему пороговому значению (критерию разбиения) узла в дереве.

Стоит также упомянуть, что в XGBoost поддерживается выбор линейных моделей в качестве базовых, но, в целом, они работают хуже деревьев.

Дополнительные оптимизации в XGBoost

Weighted quantile sketch

В случае работы с большими наборами данных в XGBoost используется приближённый жадный алгоритм. Это означает, что вместо тестирования всех возможных пороговых значений мы тестируем только квантили. Используя распараллеливание для разделения набора данных и создания ветвей в каждом дереве независимо друг от друга чтобы несколько ядер могли работать с ними одновременно, алгоритм взвешенных квантилей (weighted quantile sketch) объединяет данные в приближённую гистограмму, разделённую на взвешенные квантили, которые помещают наблюдения с прогнозами низкой достоверности в квантили с меньшим количеством наблюдений.

Cache-aware access & out-of-core computing

Для более быстрого вычисления прогнозов градиенты и гессианы хранятся в кэш-памяти (cache-aware access), а в случае с датасетами большого размера, когда недостаточно кэш и оперативной памяти, задействуется память из жёсткого диска. Учитывая, что чтение данных на жёстком диске выполняется гораздо медленнее чем в кэш-памяти, для ускоренной обработки производится их компрессия. Данные хранятся в единицах памяти, называемых блоками, информация в которых хранится в формате сжатых столбцов, отсортированных по соответствующим значениям параллельно с использованием всех доступных ядер ЦП, а распаковка каждого блока осуществляется в независимом потоке при загрузке в память. Данная техника называется "вычисление вне ядра" (out-of-core computing). При наличии нескольких жёстких дисков, применяется параллельное сегментирование блоков для каждого жёсткого диска (sharding).

При использовании сжатых данных необходимо учитывать, что процесс сжатия и распаковки занимает дополнительное время, поэтому выбор алгоритма сжатия (с потерями и без) и его параметров может повлиять на производительность в целом, однако это всё равно значительно быстрее чтения без сжатия.

Sparsity-aware split finding

В XGboost также поддерживается работа с пропусками. Суть заключается в следующем: из исходного датасета убираются наблюдения с пропусками и представляются как отдельный датасет. После построения дерева для наблюдений без пропусков рассчитываются их остатки, далее — остатки для наблюдений с пропусками, которые присоединяются к остаткам дерева как для левого, так и для правого узла. В конце рассчитывается информационный прирост (gain) для каждого узла в дереве: выбирается порог для разбиения с максимальным приростом. Данная техника называется "поиск разбиения с учётом разряженности" (sparsity-aware split finding).

Обучение на GPU

Для ускорения процесса обучения в XGBoost реализована поддержка GPU с помощью параметров device="cuda" и tree_method="gpu_hist". Также стоит отметить, что в данном алгоритме поддерживается полностью распределённое обучение на GPU с использованием Dask, Spark и PySpark.

Пакетная реализация XGBoost и оценка полученных результатов

Благодаря вышеуказанным оптимизациям и особенностям XGBoost работает лучше классической реализации, что видно на примере классификации, однако на примере регрессии в данном случае результат оказался хуже. Если хорошо подготовить данные и поэкспериментировать с настройкой гиперпараметров, XGBoost будет работать гораздо лучше как в случае регрессии, так и классификации.

XGBClassifier

from xgboost import XGBClassifier, XGBRegressor

xgbc = XGBClassifier(seed=0)
xgbc.fit(X1_train, y1_train)
xgbc_pred_res = xgbc.predict(X1_test)
xgbc_accuracy = accuracy_score(xgbc_pred_res, y1_test)
print(f'xgbc accuracy: {xgbc_accuracy}')
print(xgbc_pred_res)

feature_indexes = [1, 3]
title2 = 'XGBClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, xgbc, feature_indexes, title2)


xgbc accuracy: 0.6851851851851852
[5 0 1 4 1 1 0 1 1 1 3 0 1 1 1 5 2 1 1 2 3 0 5 5 0 0 5 0 0 1 0 0 1 0 0 0 0
 0 0 5 1 4 1 0 1 1 0 1 0 1 0 1 5 0]

XGBRegressor

xgbr = XGBRegressor(seed=0)
xgbr.fit(X2_train, y2_train)
xgbr_pred_res = xgbr.predict(X2_test)
xgbr_mape = mean_absolute_percentage_error(xgbr_pred_res, y2_test)
print(f'xgbr mape: {xgbr_mape}')
print(xgbr_pred_res)


xgbr mape: 0.3781388070785457
[281.3304   241.21165  212.13124  136.89372  144.54208  261.7871
 125.257454 236.16089  199.79964  250.18103  188.50882  126.724464
 161.2112    73.98386  302.50272   97.35103  160.85774   67.10394
  96.0406   284.72873  181.48717  117.764626 182.91393  127.76686
 229.30229  173.58592  170.03932   76.73085  240.7698   117.70699
 224.36049   63.628925 124.27327  142.07358  173.09418  139.11557
 163.61208   92.8561    95.91142  217.65016  112.03522  122.800545
 110.06953  181.34201  159.39062   61.97588  114.534805 145.20259
  84.905815 284.77567  195.29976   45.550606  92.54941  180.4689
 279.12097  241.1343   177.77785  130.43887  159.53189  158.2313
 253.25339  181.86093  120.71679   90.604706 274.5059   140.21674
  92.80314  211.71936  233.20622  110.791824  67.89235  133.62137
 123.05923  102.70884  167.2115   159.32262  131.7305   243.41493
 254.97531  210.0554   130.00981  222.21094   52.864147 251.23187
 118.602684  87.89112  121.147095 218.15724   90.9274    91.69777
 101.95428   85.79966   79.59963  155.879    118.25745   99.5975
 231.70824  189.6652   121.97507  147.51166  152.10051  102.239395
 177.74615  107.55845  271.80927  121.41044  232.97308  292.89096
 129.39447   60.501278 259.46295 ]

Преимущества и недостатки XGBoost

Преимущества:

относительно высокая скорость обучения и точность прогнозов;
возможность работы с пропусками;
гибкость в конфигурации из-за большого количества гиперпараметров.

Недостатки:

сложность в конфигурации из-за большого количества гиперпараметров;
на данный момент отсутствует полноценная (присутствует экспериментальная) возможность работы с категориальными признаками, что может сильно снизить производительность при кодировании больших объёмов данных.

Дополнительные источники

Статья «XGBoost: A Scalable Tree Boosting System», Tianqi Chen, Carlos Guestrin.

Документация.

Лекции:

один;
два.

Пошаговое построение XGBoost:

Особенности CatBoost

CatBoost (Categorical Boosting) — более эффективная реализация градиентного бустинга от компании Яндекс, особенностью которой является возможность работы с категориальными признаками без их предварительной обработки. Разберёмся подробнее как это происходит.

Ordered target (CatBoost) encoding

При кодировании категориальных признаков к бустингам применяется один из нескольких классических подходов:

target encoding;
one-hot encoding (хорошо работает лишь в том случае, когда в признаках содержится небольшое количество значений);
feature hashing в несколько корзин (исходные данные преобразуются в уникальную последовательность символов (хэш-значения), которые разбиваются на группы (корзины) и обрабатываются параллельно с последующим применением target или one-hot encoding);
mean target encoding (рассчитывается новое значение для текущего категориального признака как среднее значение целевой переменной по датасету);
leave-one-out (рассчитывается среднее значение целевой переменной по всем объектам кроме текущего);
leave-bucket-out (рассчитывается среднее значение целевой переменной на одной части данных и применяется для оставшихся других);

Однако все вышеперечисленные способы так или иначе приводят к утечке данных и переобучению, и данную проблему можно частично решить с применением mean target encoding на отложенной выборке, что работает заметно лучше, но такой способ не идеален так как остаётся меньше данных для обучения и вычисления статистик по категориальным признакам.

В CatBoost применяется модифицированный подход к mean target encoding, называемый Ordered target (CatBoost) encoding, который состоит из 3-х нижеперечисленных этапов.

Случайная перестановка

Образцы в наборе данных перемешиваются случайным образом, что позволяет создать более сбалансированный набор данных и снизить переобучение.

Статистики «по прошлому»

После случайной перестановки рассчитывается новое значение для текущего категориального признака следующим образом:

$CTR = \frac{TargetSum + Prior}{TotalCount + 1}$

Prior — константа, определяемая как среднее арифметическое целевых значений.

TargetSum — сумма целевых значений для конкретного категориального признака до текущего момента.

TotalCount — количество всех наблюдений для конкретного категориального признака до текущего момента.

Данная формула может быть применима как для регрессии, так и для мультиклассификации.

Пошаговый пример с нуля для мультиклассификации

!pip install category_encoders

from category_encoders.cat_boost import CatBoostEncoder

train = pd.DataFrame({"color": ["red", "blue", "black", "black", "blue", "blue", "red", "blue"]})
target = pd.Series([2, 1, 2, 1, 0, 3, 0, 1])

cbe_encoder = CatBoostEncoder()
train_cbe = cbe_encoder.fit_transform(train, target)
prior = target.mean()

print(f'prior: {prior}', '', sep='\n')
print(train_cbe)


prior: 1.25

    color
0  1.2500
1  1.2500
2  1.2500
3  1.6250
4  1.1250
5  0.7500
6  1.6250
7  1.3125

В случае c числовыми признаками создаётся набор меток с помощью квантования и производится их бинаризация на k классов: все значения до границы (метки) относятся к классу 0, а после к классу 1. Далее процедура схожа со случаем выше.

В CatBoost поддерживаются следующие виды квантования:

мода;
медианная сетка;
равномерная сетка + квантили (размер квантования уменьшается вдвое и разбиения (пороговые значения) объединяются по одному из двух способов: медиана или равномерная сетка);
MaxLogSum — максимизируется значение следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} log \ w_i$

MinEntropy — минимизируется значение следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} w_i \ log \ w_i$

GreedyLogSum — максимизируется аппроксимация жадным способом следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} log \ w_i$

где n — число различных объектов в сегменте, w (weight) — число повторений объекта в сегменте.

Комбинация категориальных признаков

В CatBoost используется комбинация категориальных признаков (например, красный диван, синий диван, синий стул и так далее), что позволяет модели учитывать неочевидные взаимосвязи между ними и улучшить качество итоговых прогнозов в целом. Для избегания экспоненциального роста комбинаций категориальных признаков, их создание происходит не перед началом обучения, а в момент построения каждого дерева жадным алгоритмом отдельно.

Структура решающих деревьев и упорядоченный бустинг

В качестве базовых моделей в CatBoost используются симметричные односторонние деревья, имеющие одинаковую глубину и ширину. Создатели алгоритма считают, что такой подход позволяет ускорить обучение и создавать деревья, которые менее подвержены переобучению.

Для нахождения наилучшего критерия разбиения в узле используется косинусное сходство — его максимальное значение будет соответствовать лучшему критерию:

$Cosine \ Similarity = \frac{ \sum\limits_{i=1}^{n} A_i B_i}{\sqrt{\sum\limits_{i=1}^{n} A_i^2} \sqrt{\sum\limits_{i=1}^{n} B_i^2}}$

где A — вектор с остаточными ошибками, B — вектор с прогнозами в листовых узлах.

Проблема классического бустинга заключается в том, что прогнозы базовых моделей могут иметь высокое смещение, что в конечном счёте приводит к переобучению модели в целом — это обусловлено тем, что обучение и оценка деревьев производится на одинаковом наборе данных (X_train). Данная проблема решается с помощью метода линейной аппроксимации упорядоченного бустинга (Ordered Boosting), суть которого заключается в следующем: данные перемешиваются случайным образом и после чего текущее дерево обучается на m-1 образцах с учётом остатков предыдущего, что очень похоже на принцип в ordered target encoding, то есть изначально берётся один образец из набора данных и модель учится на всех образцах до него, то есть на нулях, далее берётся два образца и модель учится на одном, после чего рассчитываются остатки, которые будут использованы для следующей модели и так далее. Такой подход позволяет произвести обучение дерева со сложностью O(log(m)) вместо O(m^2) .

Структура симметричного одностороннего дерева

Дополнительные возможности CatBoost

Работа с пропусками

Поддерживаются следующие режимы обработки пропусков:

"Forbidden" — пропуски интерпретируются как ошибки.
"Min" — пропуски обрабатываются как минимальное значение (меньшее, чем все остальные значения) для объекта.
"Max" — пропуски обрабатываются как максимальное значение (превышающее все остальные значения) для объекта.

Режим обработки по умолчанию — Min. Также при выборе деревьев учитывается разбиение, отделяющее пропущенные значения от всех остальных.

Snapshots

Данный параметр (save_snapshot=True) позволяет создавать резервные копии промежуточных результатов. Если происходит неожиданное прерывание обучения (например, случайное выключение компьютера), то оно может быть продолжено из сохранённого состояния и в таком случае не нужно повторять завершённые итерации построения деревьев.

Обучение на GPU

В CatBoost возможно обучение не только на нескольких GPU, но и сразу на нескольких серверах с GPU, что фактически позволяет запустить распределённое обучение и использовать датасеты любого размера. Данная процедура позволяет значительно ускорить работу алгоритма: чем больше датасет, тем больше будет прирост в производительности в сравнении с обучением на CPU. Для того чтобы этим воспользоваться достаточно указать task_type="GPU".

Пакетная реализация CatBoost и оценка полученных результатов

Как можно заметить из примера классификации, не смотря на большое количество улучшений, CatBoost далеко не всегда показывает высокие результаты в сравнении с другими реализациями градиентного бустинга, как это заявляют его разработчики, особенно заметна разница в случае с несбалансированными наборами данных. Однако, стоит отметить, что в случае хорошей подготовки данных, при прочих равных, в ряде случаев CatBoost будет отличаться от других модификаций немного более высокой точностью.

В случае регрессии, когда нет явно выраженной проблемы как в случае выше, CatBoost справляется довольно хорошо.

Полученные результаты приведены ниже.

!pip install catboost

CatBoostClassifier

from catboost import CatBoostClassifier, CatBoostRegressor

cb_clf = CatBoostClassifier(random_seed=0, logging_level='Silent')
cb_clf.fit(X1_train, y1_train)
cb_clf_pred_res = cb_clf.predict(X1_test).squeeze()   # remove all single-dimensional entries
cb_clf_accuracy = accuracy_score(cb_clf_pred_res, y1_test)
print(f'cb_clf accuracy: {cb_clf_accuracy}')
print(cb_clf_pred_res)

feature_indexes = [1, 3]
title3 = 'CatBoostClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, cb_clf, feature_indexes, title3)


cb_clf accuracy: 0.6851851851851852
[5 0 1 4 1 1 0 1 1 1 0 0 1 1 1 5 2 1 2 2 1 0 5 1 0 0 5 0 1 1 0 0 1 0 0 0 0
 0 0 5 1 5 1 0 1 1 0 1 0 1 0 1 5 0]

CatBoostRegressor

cb_reg = CatBoostRegressor(random_seed=0, logging_level='Silent')
cb_reg.fit(X2_train, y2_train)
cb_reg_pred_res = cb_reg.predict(X2_test)
cb_reg_mape = mean_absolute_percentage_error(cb_reg_pred_res, y2_test)
print(f'cb_reg mape: {cb_reg_mape}')
print(cb_reg_pred_res)


cb_reg mape: 0.3260573594870776
[268.21698587 226.61112865 152.56164473 105.39534082 185.56084388
 231.13386917 112.94031582 246.11217136 121.83631454 238.66069067
 193.92810529 171.53207065 120.95427566  98.87431921 301.0159263
  97.989809   170.68360243  70.03044356 104.9534818  235.1624283
 174.7895105  131.39173163 178.28683674 128.13631321 230.60611298
 186.37090017 131.52230822  61.453026   231.95028492 148.67901568
 186.2317954   80.45908918 137.68617262 171.24445717 128.77648933
 178.31364898 179.17236177 149.82094832 102.25676058 209.48231728
 116.01443961 169.06621701 110.9047463  210.24240477 158.28102362
  72.4532758  135.86931928 121.14865532  76.8103627  278.66222581
 127.85077736  55.25704176 144.51194393 170.06468442 271.51905928
 187.61879537 159.56680649 118.65520621 121.92004997 183.56316238
 248.87847858 137.77180529 117.98743897  82.53109543 255.59388802
 151.23062005  94.00606836 257.15481144 227.21705762  96.77322241
  78.34198817 118.09701594  91.2069645   91.70192792 125.65538331
 155.68222799 129.63411563 230.46496307 292.38275237 213.68057983
 131.76653094 217.93105223  50.37188156 256.16334997  94.4107044
  90.95547399 139.58356594 187.26517    127.97009716 171.85831366
  89.65051176  95.93129442  60.14437246 178.27466698 101.36657382
 106.15753799 206.11327567 217.74079838 130.83283251 138.63144499
 154.18156801 123.38315127 197.65233246 125.95249034 239.25024631
 139.69849648 245.02582928 288.17892899 122.76437963  70.023686
 233.36946028]

Преимущества и недостатки CatBoost

Преимущества:

высокая точность и скорость обучения в большинстве случаев;
возможность работы с категориальными признаками;
большое количество гиперпараметров для настройки алгоритма;
применяется для многих задач.

Недостатки:

нестабильная работа с несбалансированными данными;
сложность в конфигурации из-за большого количества гиперпараметров;
плохая документация (личное мнение).

Дополнительные источники

Статья «CatBoost: unbiased boosting with categorical features», Liudmila Prokhorenkova, Gleb Gusev, Aleksandr Vorobev, Anna Veronika Dorogush, Andrey Gulin.

Документация.

Лекция.

Пошаговое построение CatBoost (в ordered target encoding есть ошибки):

один;
два.

Особенности LightGBM

LightGBM (Light Gradient Boosting Machine) — реализация градиентного бустинга от компании Microsoft, способная обрабатывать большие наборы данных и при этом обеспечивать высокую точность. Рассмотрим более подробно ключевые особенности, которые на это повлияли.

Gradient-based One-Side Sampling (GOSS)

В отличие от классического градиентного бустинга, где регрессионные деревья обучаются на всём множестве X_train, в LightGBM деревья обучаются на подмножестве данных, полученном с помощью одностороннего отбора на основе градиента (GOSS), суть которого заключается в сортировке и выборе образцов top N с наибольшим градиентом — это означает, что именно этим данным следует уделить особое внимание, поскольку на них дерево будет ошибаться больше всего. Помимо этого, случайным образом производится отбор части образцов с низким градиентом с последующим присвоением им весов $\frac{1-a}{b}$ , чтобы избежать изменений в распределении данных. В данном случае — доля образцов top N в процентах с наибольшим градиентом, — доля образцов в процентах с низким градиентом.

Проще говоря, использование GOSS позволяет значительно ускорить обучение каждого последующего дерева в ансамбле путём уменьшения обучающего набора данных без особой потери в качестве прогнозов в большинстве случаев.

Histogram-based tree node splitting

При поиске наилучшего критерия разбиения узла в LightGBM также можно использовать гистограмму значений признаков: перебираются все пороговые значения гистограммы вместо всех значений признаков, что позволяет существенно ускорить поиск наилучшего критерия. Например, для признака "рост" можно использовать вопросы "рост" ⩽ 160, "рост" ⩽ 170 вместо "рост" ⩽ 163, "рост" ⩽ 167 и так далее. Как можно было заметить, гистограммы применяются в том или ином виде во всех трёх модификациях градиентного бустинга.

Exclusive Feature Bundling (EFB)

Предположим, что тем или иным образом в датасете имеются закодированные категориальные признаки, большинство из которых являются взаимоисключающими (те, которые одновременно не принимают ненулевые значения). При большом числе таких признаков обучение модели может занять слишком много времени и для решения данной проблемы в LightGBM применяется понижение размерности пространства признаков путём объединения взаимоисключающих признаков (EFB).

Стоит отметить, что в действительности строго взаимоисключающие признаки в данных встречаются крайне редко. Поэтому вводится такое понятие как доля конфликтов между признаками — отношение наблюдений от общего числа, где два признака принимают одновременно ненулевые значения.

Допустим, имеется датасет с признаками F0−F4 . В первую очередь стоит установить допустимую долю конфликтов, например, 25%. Тогда в случае из 8 образцов допустимое число конфликтов будет составлять 2. Далее строится матрица конфликтов между всеми признаками и производится подсчёт общей суммы конфликтов для каждого признака, после чего происходит их сортировка в порядке убывания.

На основе матрицы конфликтов строится взвешенный граф, у которого в полученном порядке отсекаются все рёбра с числом конфликтов, превышающим допустимое значение.

Работа Exclusive Feature Bundling с нуля на простом примере

Первой связкой будет F_0−F_1 . Не смотря на то что F_1−F_2 и F_1−F_3 не конфликтуют, их объединение невозможно из-за конфликтов F_0−F_2 и F_0−F_3 . Следующей связкой будет F_2−F_4 . Не смотря на то, что F_4 и F_3 не конфликтуют, их объединение невозможно из-за конфликта F_2−F_3 . Последним остаётся признак F_3 .

Новая связка будет состоять из ненулевых значений первого признака и ненулевых значений второго плюс смещение, равное максимальному значению первого признака: смещение каждого следующего признака будет равно сумме максимальных значений предыдущих. В случаях, когда наблюдается конфликт, берётся значение последнего признака со смещением.

Подробные расчёты приведены ниже.

Нетрудно заметить, что алгоритм EFB можно оптимизировать, игнорируя вычисления для всех нулевых значений в признаках. Также данную оптимизацию можно применить к histogram-based tree node splitting, что в целом снизит сложность построения гистограммы с O(data) до O(non-zero data).

Leaf-wise tree growth

Вместо построения деревьев слой за слоем, в LightGBM деревья строятся в глубину: рост продолжается в листовом узле с наибольшим информационным приростом, в результате чего дерево получается глубоким и несимметричным. Чаще всего такой подход превосходит симметричные деревья в плане точности.

Обучение на GPU

При работе с большими датасетами в LightGBM поддерживается обучение не только на GPU, но и распределённое обучение сразу на нескольких машинах.

На данный момент существуют следующие подходы:

Data parallel (образцы датасета разбиваются на локальные гистограммы, каждая из которых обрабатывается отдельно);
Feature parallel (признаки датасета разбиваются на локальные гистограммы, каждая из которых обрабатывается отдельно);
Voting parallel (на каждом процессе локально выбирается K лучших признаков для разбиения, после чего выбирается наилучший из них глобально).

Пакетная реализация LightGBM и оценка полученных результатов

Как можно заметить, в случае классификации LightGBM показал результат из коробки значительно лучше, чем предыдущие модификации бустинга. Также стоит отметить, что из всех трёх рассмотренных модификацией градиентного бустинга LightGBM является самой быстрой: чем больше датасет, тем сильнее будет заметна разница в плане скорости.

Полученные результаты приведены ниже.

LGBMClassifier

from lightgbm import LGBMClassifier, LGBMRegressor

lgb_clf = LGBMClassifier(random_state=0, verbose=-1)
lgb_clf.fit(X1_train, y1_train)
lgb_clf_pred_res = lgb_clf.predict(X1_test)
lgb_clf_accuracy = accuracy_score(lgb_clf_pred_res, y1_test)
print(f'lgb_clf accuracy: {lgb_clf_accuracy}')
print(lgb_clf_pred_res)

feature_indexes = [1, 3]
title4 = 'LGBMClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, lgb_clf, feature_indexes, title4)


lgb_clf accuracy: 0.7407407407407407
[5 0 1 4 1 1 0 1 1 1 0 0 0 1 1 5 2 1 2 2 3 1 5 5 0 0 5 0 1 1 0 0 1 0 0 0 0
 0 0 5 1 4 1 0 0 1 0 1 0 1 0 1 5 0]

LGBMRegressor

lgb_reg = LGBMRegressor(random_state=0, verbose=-1)
lgb_reg.fit(X2_train, y2_train)
lgb_reg_pred_res = lgb_reg.predict(X2_test)
lgb_reg_mape = mean_absolute_percentage_error(lgb_reg_pred_res, y2_test)
print(f'lgb_reg_mape: {lgb_reg_mape}')
print(lgb_reg_pred_res)


lgb_reg_mape: 0.349719771087187
[261.46864967 223.24674049 179.60231224 104.76984808 197.67529058
 264.26116883 111.11355079 243.72059654 144.18045853 239.08131542
 199.90841705 170.37417433 151.67933516  89.30779664 297.93561211
  96.58862229 162.53285879  70.06113305 108.36436227 234.16276627
 207.7552967  136.83913826 172.38724529 110.26981866 228.1574526
 163.56817258 108.94516511  73.93303203 233.31869769 116.00042252
 195.39846082  72.54403358 164.38943926 194.42418913 130.42154381
 183.04435501 173.56352142 147.00774306  89.07365731 175.25759644
 108.72749406 144.48000593  99.60713604 196.86889512 160.47835096
  60.17798032 133.4909559  115.46543242  64.48286456 251.20520649
 135.52608509  45.16396653 161.70420785 187.28406745 274.21158331
 179.30355071 150.1563969  111.1275583  100.5067047  133.7868607
 269.98464504 159.03980859 118.01578638  75.41625416 249.0569831
 133.99026562 111.43954756 221.81954307 206.02676942  81.56098377
  72.5920367  101.31664062  93.31261471 111.24487763 149.40048547
 177.37205369 130.08118649 212.0573     294.138509   232.24330644
 109.5457663  181.3487432   63.20386576 261.47508264  88.43007908
  99.59568927 127.97978786 211.77672798 121.44351586 138.27282283
 121.01455597  91.26965383  68.92870597 156.03234199 101.80734511
 108.60244399 214.13968395 218.58499151 131.09438025 114.99269899
 187.63688894 117.5399017  165.85959666 110.74981366 238.97897095
 114.01806217 240.07138204 283.0195311  108.08395213  67.44250127
 238.97271264]

Преимущества и недостатки LightGBM

Преимущества:

высокая точность прогнозов и особенно скорость обучения;
возможность работы с категориальными признаками;
большое количество гиперпараметров для настройки алгоритма;
применяется для многих задач.

Недостатки:

возможна нестабильная работа на датасетах небольшого размера;
сложность в конфигурации из-за большого количества гиперпараметров.

Дополнительные источники

Статья «LightGBM: A Highly Efficient Gradient Boosting Decision Tree», Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, Tie-Yan Liu.

Документация.

Лекции:

Какой бустинг лучше

Однозначного ответа на вопрос какой бустинг лучше нет. Если раньше можно было сказать, что при работе с категориальными признаками лучше использовать CatBoost, когда важна скорость — LightGBM, ну или XGBoost, если хочется стабильное решение и нет времени на эксперименты, то сейчас такое утверждение будет неверным: как можно было заметить, все три модификации успели скопировать друг у друга много хороших идей, но исторически они отличались довольно сильно.

Более правильным будет утверждение, что выбор наилучшего бустинга будет зависеть от конкретной задачи и данных, которые применяются для обучения модели: в разных сценариях одни и те же алгоритмы могут показывать как лучшие результаты в плане скорости и точности, так и наоборот — худшие. Таким образом, выбор наиболее оптимального бустинга будет заключаться в проведении экспериментов с разными модификациями и сравнении полученных результатов.

Несколько полезных источников со сравнением бустингов: один, два, три.

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13 мар. 2024 г.

11.09.25 15:03 stevensjonas

Recovery of Stolen Ethereum Wallet. Hire: Supreme Peregrine Recovery. I would like to express my sincere gratitude to Supreme Peregrine Recovery for their outstanding help in recovering my stolen Ethereum wallet, which held $594,684 worth of assets. I thought my money was lost forever after falling for a scam that imitated a reliable cryptocurrency platform, but I was referred to them because their staff was professional, knowledgeable, and genuinely concerned from the start. Their commitment, openness, and expertise are unparalleled. I am immensely appreciative of their assistance and heartily recommend Supreme Peregrine Recovery to anyone dealing with cryptocurrency theft. They transformed a nightmare into a tale of hope and recovery. They carefully examined my case, gave me clear updates at every stage, and used cutting-edge Blockchain tracing tools to recover the entire amount. +1,3,1,8,5,5,3,0,6,7,9 Mail: supremeperegrinerecovery(@)proton(.)me supremeperegrinerecovery567(@)zohomail(.)com info(@)supremeperegrinerecovery(.)com
11.09.25 18:51 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.09.25 18:51 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.09.25 18:51 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.09.25 13:11 grace

I lost my money to them few months ago. I lost over $244,000, they denied my withdrawal request and and also left it pending. I reached out to them and they never responded back tooth my emails and calls. They eventually locked me out of my account. I had to reach out to a recovery expert ( anthonydaviestech@gmail com) to help me recover all my money back. I have gotten my money back**you can also text him via whatsapp: +19514908435
12.09.25 15:06 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.09.25 15:06 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.09.25 20:45 Clinton

I know firsthand how devastating it feels to lose money to a scam. When it happened to me, I was left frustrated, hopeless, and unsure where to turn. Complaining to apps about being scammed on their platforms didn’t change anything it felt like no one cared. But through a friend’s advice, I connected with a recovery professional who gave me real hope. With their help, I was able to get my funds back, something I thought was impossible. I’m sharing this because I don’t want anyone else to feel as lost as I once did. Whether you’ve been scammed through crypto platforms, dating scams, real-estate or mortgage fraud, or even fake ICOs, please know there’s still a way forward. If you’ve lost money, reach out to Recoveryfundprovider(@)gmail(.)com).) or (WhatsApp) at (+)44 7366445035 for consultation and recovery services.
14.09.25 00:54 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14.09.25 00:54 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14.09.25 02:37 luciajessy3

Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / WHATSAPP ; +1 (603) 702 ( 4335 ) is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…
14.09.25 02:37 luciajessy3

Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / WHATSAPP ; +1 (603) 702 ( 4335 ) is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…
14.09.25 12:44 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14.09.25 12:44 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14.09.25 15:13 estherfords

My solana recovery experience with Mighty Hacker Recovery Expert.
14.09.25 15:13 estherfords

My solana recovery experience with Mighty Hacker Recovery Expert. Contact them on +14042456415 WhatsApp
14.09.25 17:48 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14.09.25 17:48 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14.09.25 20:49 Clinton

I know firsthand how devastating it feels to lose money to a scam. When it happened to me, I was left frustrated, hopeless, and unsure where to turn. Complaining to apps about being scammed on their platforms didn’t change anything it felt like no one cared. But through a friend’s advice, I connected with a recovery professional who gave me real hope. With their help, I was able to get my funds back, something I thought was impossible. I’m sharing this because I don’t want anyone else to feel as lost as I once did. Whether you’ve been scammed through crypto platforms, dating scams, real-estate or mortgage fraud, or even fake ICOs, please know there’s still a way forward. If you’ve lost money, reach out to ([email protected]) or (WhatsApp) at (+)44 7366445035 for consultation and recovery services.
14.09.25 22:16 Berrysmith

I recently recovered a large sum of digital assets. Sylvester Bryant, reachable at Yt7cracker@gmail. com, helped me reclaim 720,000 in Ethereum. The entire experience was excellent. I highly recommend Mr. Sylvester. He assists those dealing with lost or stolen funds. His skill in getting back money from scams is outstanding. Scammers cause huge stress and financial pain. Mr. Bryant offers a crucial service. He provides a way to possibly recover stolen funds. His professional manner built trust. If you lost money to scams, contact him. He is also on WhatsApp at +1 512 577 7957. This makes it easy to discuss your case. Recovering this amount was difficult. Mr. Bryant showed deep knowledge of the technical work. His commitment to solving these problems is clear. Seek his help for lost crypto or other digital assets. This is a major concern for many. Expert aid can greatly improve outcomes.
15.09.25 10:25 aliceforemanlaw

I got back a lot of digital money recently. Sylvester Bryant helped me recover 120,000 in Ethereum. You can reach him at Yt7cracker@gmail. com. It was a great experience overall. I really suggest Mr. Sylvester. He helps people who lost money. He is great at getting money back from scams. Scammers cause a lot of stress and hurt. Mr. Bryant offers an important service. He can help you get back stolen funds. He was very professional and earned my trust. Contact him if you lost money to scams. He is also on WhatsApp at +1 512 577 7957. This makes it easy to talk about your situation. Getting this amount back was tough. Mr. Bryant knew a lot about the technical parts. His drive to fix these issues is clear. Get his help if you lost crypto or other digital items. Many people worry about this. Getting expert help can really make a difference.
15.09.25 12:49 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.09.25 12:49 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.09.25 12:49 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.09.25 12:25 aliceforemanlaw

Sylvester Bryant helped me recover 120,000 in Ethereum. You can reach him at Yt7cracker@gmail. com. It was a great experience overall. I really suggest Mr. Sylvester. He helps people who lost money. He is great at getting money back from scams. Scammers cause a lot of stress and hurt. Mr. Bryant offers an important service. He can help you get back stolen funds. He was very professional and earned my trust. Contact him if you lost money to scams. He is also on WhatsApp at +1 512 577 7957. This makes it easy to talk about your situation. Getting this amount back was tough. Mr. Bryant knew a lot about the technical parts. His drive to fix these issues is clear. Get his help if you lost crypto or other digital items. Many people worry about this. Getting expert help can really make a difference.
16.09.25 12:26 aliceforemanlaw

Sylvester Bryant helped me recover 120,000 in Ethereum. You can reach him at Yt7cracker@gmail. com. It was a great experience overall. I really suggest Mr. Sylvester. He helps people who lost money. He is great at getting money back from scams. Scammers cause a lot of stress and hurt. Mr. Bryant offers an important service. He can help you get back stolen funds. He was very professional and earned my trust. Contact him if you lost money to scams. He is also on WhatsApp at +1 512 577 7957. This makes it easy to talk about your situation. Getting this amount back was tough. Mr. Bryant knew a lot about the technical parts. His drive to fix these issues is clear. Get his help if you lost crypto or other digital items. Many people worry about this. Getting expert help can really make a difference.
16.09.25 13:25 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.09.25 13:25 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.09.25 02:56 peju1213

It can be very concerning to lose access to your USDT (Tether) wallet. It could potentially result in permanent loss of significant funds. You may have misplaced your private keys. Perhaps you unintentionally erased something, or your security was breached. It seems hopeless when you are unable to access your USDT. However, it doesn't have to stop there. There are methods for recovering your lost USDT. During difficult circumstances, professional assistance and wise recuperation techniques can give you hope. get in touch with Marie ([email protected] and +1 7127594675) on WhatsApp.
17.09.25 11:16 [email protected]

Sylvester Bryant helped me recover $510,000 in Ethereum, and I highly recommend his services. He specializes in helping people recover funds lost to scams. Scammers can cause significant stress, but Mr. Bryant's professionalism and expertise in the technical aspects of recovery earned my trust. If you've lost money through scams, you can reach him at [ yt7cracker@gmail. com ] or on WhatsApp at +1 512 577 7957. His dedication to helping people get their stolen funds back is clear, and his support made a huge difference in my case.
17.09.25 11:16 [email protected]

Sylvester Bryant helped me recover $510,000 in Ethereum, and I highly recommend his services. He specializes in helping people recover funds lost to scams. Scammers can cause significant stress, but Mr. Bryant's professionalism and expertise in the technical aspects of recovery earned my trust. If you've lost money through scams, you can reach him at [ yt7cracker@gmail. com ] or on WhatsApp at +1 512 577 7957. His dedication to helping people get their stolen funds back is clear, and his support made a huge difference in my case.
17.09.25 23:50 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.09.25 23:50 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.09.25 23:50 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.09.25 11:55 Mundo

Sylvester Bryant recovered £400,000 in Ethereum for me. I strongly advise using his services. He focuses on retrieving money lost to fraudulent schemes. Scams cause immense worry. Mr. Bryant's expert knowledge and calm approach in getting funds back reassured me. If you have lost money to scams, contact him via email at [email protected]. You can also reach him on WhatsApp at +1 512 577 7957. He is committed to helping people recover their stolen assets. His support was invaluable in my situation.
18.09.25 11:55 Mundo

Sylvester Bryant recovered £400,000 in Ethereum for me. I strongly advise using his services. He focuses on retrieving money lost to fraudulent schemes. Scams cause immense worry. Mr. Bryant's expert knowledge and calm approach in getting funds back reassured me. If you have lost money to scams, contact him via email at [email protected]. You can also reach him on WhatsApp at +1 512 577 7957. He is committed to helping people recover their stolen assets. His support was invaluable in my situation.
18.09.25 12:22 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.09.25 12:22 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.09.25 13:04 Mundo

Sylvester Bryant recovered £400,000 in Ethereum for me. I strongly advise using his services. He focuses on retrieving money lost to fraudulent schemes. Scams cause immense worry. Mr. Bryant's expert knowledge and calm approach in getting funds back reassured me. If you have lost money to scams, contact him via email at [email protected]. You can also reach him on WhatsApp at +1 512 577 7957. He is committed to helping people recover their stolen assets. His support was invaluable in my situation.
18.09.25 16:46 carolinehudso83

I noticed a lot of online recommendations, and it’s clear that some of them are bad eggs that will just make уour mуsterу worse. I can onlу suggest one, and if уou need assistance getting back the moneу уou lost to scammers, уou can contact them bу email at: [email protected] Whats app or text +1 518 468 2985 Website https://bestrecoveryagent.com/
19.09.25 00:54 frank2025

Anthony davies played a crucial role in successfully recovering my stolen USDT, which was valued at more than $300,000. His expertise and dedication made the recovery process smooth and efficient. Throughout the entire time, he demonstrated himself to be a reliable recovery agent. His professionalism and commitment to his clients are evident in the results he achieves. If you find yourself in a similar situation or need assistance, he is available for contact. You can reach anthony through email at (anthonydaviestech AT gmail dot com). Alternatively, he is also accessible via telegram at anthonydavies01. Reaching out to him could be the first step toward recovering your lost assets
19.09.25 10:02 faithlawrence

Sylvester Bryant recovered £400,000 in Ethereum for me. I strongly advise using his services. He focuses on retrieving money lost to fraudulent schemes. Scams cause immense worry. Mr. Bryant's expert knowledge and calm approach in getting funds back reassured me. If you have lost money to scams, contact him via email at [email protected]. You can also reach him on WhatsApp at +1 512 577 7957. He is committed to helping people recover their stolen assets. His support was invaluable in my situation.
19.09.25 10:02 faithlawrence

Sylvester Bryant recovered £400,000 in Ethereum for me. I strongly advise using his services. He focuses on retrieving money lost to fraudulent schemes. Scams cause immense worry. Mr. Bryant's expert knowledge and calm approach in getting funds back reassured me. If you have lost money to scams, contact him via email at [email protected]. You can also reach him on WhatsApp at +1 512 577 7957. He is committed to helping people recover their stolen assets. His support was invaluable in my situation.
19.09.25 15:27 tonykith01477

There are not many reviews about free crypto recovery fixed but the ones that exist are mostly positive. I have no idea how that is possible, but it is clear that they are fabricated. However, there are also negative reviews – and I highly recommend paying attention to them. These are real people describing their unfortunate experiences with this scam. It’s the same story as always – fraudsters create a fake website to con people out of money. Nothing new but look no more and contact FASTRECOVERYAGENT These types of scams are everywhere. If you have already lost money to scammers i mean any type be it sending to scammers account , or if you have a case about stolen bitcoin , usdt, or any type of cryptocurrency whatsoever just reach out to www.fastrecoveryagent.com, i really owe this group of recovery expert because they saved me after bad investment with a fake broker, i literally owe them my life as they saved me from some bunch of scammers, but see how life is , i got every cent of my investment back with the profit , reach out to to them today and tell them from Mary because i promise them i will tell the whole world when the recovery is complete, and here i go after i got my recovered assets thats why im doing what i promised. do not fall for another scam. contact the,m today!!!!
19.09.25 17:12 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19.09.25 17:12 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
20.09.25 15:05 faithlawrence

I can’t recommend Sylvester Bryant enough. He successfully recovered €210,000 in Ethereum for me after I had lost hope. His expertise in tracing and reclaiming funds from scams is truly remarkable. What stood out most was his calm, professional approach and deep technical knowledge, which gave me complete confidence throughout the process. If you’ve lost money to a scam, I strongly encourage you to reach out to him. You can contact him via email at [[email protected] ] or on WhatsApp at +1 512 577 7957. He is genuinely dedicated to helping people recover what they’ve lost, and his support has been invaluable to me.
20.09.25 15:53 blessing1198

It is distressing to lose USDT due to a bitcoin wallet hack. Although it is challenging, stolen USDT can be recovered. Taking prompt, wise action increases your chances. Marie can help you with reporting the theft, what to do right away, and USDT recovery. Contact her on WhatsApp at +1 7127594675, or email [email protected].
21.09.25 12:05 star1121

Consider this: In cryptocurrency, you see what appears to be a fantastic opportunity. After investing your hard-earned money, you see it disappear into the wallet of a con artist. Your bank account seems empty as rage and regret merge in this stomach punch. These days, cryptocurrency frauds are very common, however recovering your cryptocurrency from scammers is still possible. You can track down and possibly recover what you lost with shrewd actions. Marie can help; contact her at [email protected] and on WhatsApp at +1 7127594675.
23.09.25 03:23 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.09.25 03:23 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.09.25 03:23 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.09.25 03:54 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.09.25 14:45 Raymondkrisitina

Sylvester Bryant gets my full recommendation. He helped me get back €230,000 in Ethereum when I thought it was gone forever. His skills in tracking and retrieving scam-lost funds amaze me. I appreciated his steady, expert manner and strong tech know-how. That built my trust every step of the way. If a scam has taken your money, get in touch with him right away. Email him at yt7cracker@gmail. com or message on WhatsApp at +1 512 577 7957. He truly commits to aiding folks in regaining their assets. His help meant everything to me.
23.09.25 19:27 beverlyalex010

In 2022, I lost 1.5 ETH in what I thought was a promising Twitter investment. I reached out to the so-called brokers behind it, but they vanished and stopped replying. For years, I carried the heavy belief that my crypto was gone forever. Then, by chance, I discovered RecoveryFundProvider [email protected] WhatsApp at +44 736-644-5035 through a finance podcast. That moment changed everything. Their team didn’t lure me in with false promises or flashy guarantees. Instead, they explained their process clearly specializing in advanced blockchain tracing, smart contract forensics, and even transaction reversal in certain cases. Skeptical but hopeful, I allowed them to review my case. What followed was beyond my expectations. They dug deep into the blockchain ledger, followed the coin’s complex movements across multiple addresses, and even collaborated with global nodes to trace every trail. Two weeks later, I received an alert I never thought I’d see again: my crypto had been fully recovered and safely transferred back to me. I cried tears of relief. What sets RecoveryFundProvider apart isn’t just their cutting-edge expertise, but their unwavering ethics and dedication. They treat every client like more than just another case number even years after a loss. Today, I can confidently say they live up to their name, and I’m endlessly grateful.
23.09.25 22:38 Raymondkrisitina

Sylvester Bryant gets my full recommendation. He helped me get back €230,000 in Ethereum when I thought it was gone forever. His skills in tracking and retrieving scam-lost funds amaze me. I appreciated his steady, expert manner and strong tech know-how. That built my trust every step of the way. If a scam has taken your money, get in touch with him right away. Email him at yt7cracker@gmail. com or message on WhatsApp at +1 512 577 7957. He truly commits to aiding folks in regaining their assets. His help meant everything to me.
24.09.25 12:22 beverlyalex010

In 2022, I lost 1.5 ETH in what I thought was a promising Twitter investment. I reached out to the so-called brokers behind it, but they vanished and stopped replying. For years, I carried the heavy belief that my crypto was gone forever. Then, by chance, I discovered RecoveryFundProvider [email protected] WhatsApp at +44 736-644-5035 through a finance podcast. That moment changed everything. Their team didn’t lure me in with false promises or flashy guarantees. Instead, they explained their process clearly specializing in advanced blockchain tracing, smart contract forensics, and even transaction reversal in certain cases. Skeptical but hopeful, I allowed them to review my case. What followed was beyond my expectations. They dug deep into the blockchain ledger, followed the coin’s complex movements across multiple addresses, and even collaborated with global nodes to trace every trail. Two weeks later, I received an alert I never thought I’d see again: my crypto had been fully recovered and safely transferred back to me. I cried tears of relief. What sets RecoveryFundProvider apart isn’t just their cutting-edge expertise, but their unwavering ethics and dedication. They treat every client like more than just another case number even years after a loss. Today, I can confidently say they live up to their name, and I’m endlessly grateful.
24.09.25 15:50 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.09.25 15:50 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.09.25 15:50 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.09.25 19:08 Raymondkrisitina

Sylvester Bryant gets my full recommendation. He helped me get back €230,000 in Ethereum when I thought it was gone forever. His skills in tracking and retrieving scam-lost funds amaze me. I appreciated his steady, expert manner and strong tech know-how. That built my trust every step of the way. If a scam has taken your money, get in touch with him right away. Email him at yt7cracker@gmail. com or message on WhatsApp at +1 512 577 7957. He truly commits to aiding folks in regaining their assets. His help meant everything to me.
24.09.25 19:50 Raymondkrisitina

Sylvester Bryant gets my full recommendation. He helped me get back €230,000 in Ethereum when I thought it was gone forever. His skills in tracking and retrieving scam-lost funds amaze me. I appreciated his steady, expert manner and strong tech know-how. That built my trust every step of the way. If a scam has taken your money, get in touch with him right away. Email him at yt7cracker@gmail. com or message on WhatsApp at +1 512 577 7957. He truly commits to aiding folks in regaining their assets. His help meant everything to me.
24.09.25 19:55 Athea5130

ETHICAL BITCOIN HACKER/ BEST WAY RECOVERY BITCOIN I believed I was completely lost. I was unable to withdraw money from my investment wallet after making a dubious investment, and until I discovered Best Way Hacker Company, no one could assist me in unfreezing it. This group handled my matter with attention and urgency. In a matter of days, they were able to retrieve my frozen investment and transfer it to a new wallet that I owned. They restored my hope in a big way. Best Way Hacker restored my money and my peace of mind, and for that I will always be grateful. Email.. [email protected] WhatsApp..+39-(350)-967-(1925)
24.09.25 19:56 Athea5130

ETHICAL BITCOIN HACKER/ BEST WAY RECOVERY BITCOIN I believed I was completely lost. I was unable to withdraw money from my investment wallet after making a dubious investment, and until I discovered Best Way Hacker Company, no one could assist me in unfreezing it. This group handled my matter with attention and urgency. In a matter of days, they were able to retrieve my frozen investment and transfer it to a new wallet that I owned. They restored my hope in a big way. Best Way Hacker restored my money and my peace of mind, and for that I will always be grateful. Email.. [email protected] WhatsApp..+39-(350)-967-(1925)
24.09.25 19:56 Athea5130

ETHICAL BITCOIN HACKER/ BEST WAY RECOVERY BITCOIN I believed I was completely lost. I was unable to withdraw money from my investment wallet after making a dubious investment, and until I discovered Best Way Hacker Company, no one could assist me in unfreezing it. This group handled my matter with attention and urgency. In a matter of days, they were able to retrieve my frozen investment and transfer it to a new wallet that I owned. They restored my hope in a big way. Best Way Hacker restored my money and my peace of mind, and for that I will always be grateful. Email.. [email protected] WhatsApp..+39-(350)-967-(1925)
24.09.25 19:56 Athea5130

ETHICAL BITCOIN HACKER/ BEST WAY RECOVERY BITCOIN I believed I was completely lost. I was unable to withdraw money from my investment wallet after making a dubious investment, and until I discovered Best Way Hacker Company, no one could assist me in unfreezing it. This group handled my matter with attention and urgency. In a matter of days, they were able to retrieve my frozen investment and transfer it to a new wallet that I owned. They restored my hope in a big way. Best Way Hacker restored my money and my peace of mind, and for that I will always be grateful. Email.. [email protected] WhatsApp..+39-(350)-967-(1925)
24.09.25 20:52 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.09.25 20:52 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.09.25 12:29 beverlyalex010

In 2022, I lost 1.5 ETH in what I thought was a promising Twitter investment. I reached out to the so-called brokers behind it, but they vanished and stopped replying. For years, I carried the heavy belief that my crypto was gone forever. Then, by chance, I discovered RecoveryFundProvider (recoveryfundprovider @ gmail. com WhatsApp +44 736-644-5035) through a finance podcast. That moment changed everything. Their team didn’t lure me in with false promises or flashy guarantees. Instead, they explained their process clearly specializing in advanced blockchain tracing, smart contract forensics, and even transaction reversal in certain cases. Skeptical but hopeful, I allowed them to review my case. What followed was beyond my expectations. They dug deep into the blockchain ledger, followed the coin’s complex movements across multiple addresses, and even collaborated with global nodes to trace every trail. Two weeks later, I received an alert I never thought I’d see again: my crypto had been fully recovered and safely transferred back to me. I cried tears of relief. What sets RecoveryFundProvider apart isn’t just their cutting-edge expertise, but their unwavering ethics and dedication. They treat every client like more than just another case number even years after a loss. Today, I can confidently say they live up to their name, and i am endlessly grateful.
26.09.25 15:42 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.09.25 15:42 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.09.25 16:40 Slimbella

I fully recommend Sylvester Bryant. He recovered €230,000 in Ethereum for me after I believed it was lost for good. His talent for tracing and reclaiming funds stolen by scammers impresses me deeply. I valued his calm, skilled approach and solid grasp of technology. This gained my confidence at every turn. If scammers have stolen your cash, contact him now. Reach out by email at yt7cracker@gmail. com or via WhatsApp at +1 512 577 7957. He dedicates himself to helping people recover their belongings. His support changed everything for me.
26.09.25 16:40 Slimbella

I fully recommend Sylvester Bryant. He recovered €230,000 in Ethereum for me after I believed it was lost for good. His talent for tracing and reclaiming funds stolen by scammers impresses me deeply. I valued his calm, skilled approach and solid grasp of technology. This gained my confidence at every turn. If scammers have stolen your cash, contact him now. Reach out by email at yt7cracker@gmail. com or via WhatsApp at +1 512 577 7957. He dedicates himself to helping people recover their belongings. His support changed everything for me.
26.09.25 19:08 beverlyalex010

In 2022, I lost 1.5 ETH in what I thought was a promising Twitter investment. I reached out to the so-called brokers behind it, but they vanished and stopped replying. For years, I carried the heavy belief that my crypto was gone forever. Then, by chance, I discovered RecoveryFundProvider ( [email protected]. WhatsApp +44 736-644-5035) through a finance podcast. That moment changed everything. Their team didn’t lure me in with false promises or flashy guarantees. Instead, they explained their process clearly specializing in advanced blockchain tracing, smart contract forensics, and even transaction reversal in certain cases. Skeptical but hopeful, I allowed them to review my case. What followed was beyond my expectations. They dug deep into the blockchain ledger, followed the coin’s complex movements across multiple addresses, and even collaborated with global nodes to trace every trail. Two weeks later, I received an alert I never thought I’d see again: my crypto had been fully recovered and safely transferred back to me. I cried tears of relief. What sets RecoveryFundProvider apart isn’t just their cutting-edge expertise, but their unwavering ethics and dedication. They treat every client like more than just another case number even years after a loss. Today, I can confidently say they live up to their name, and i am endlessly grateful.
26.09.25 19:53 carolinehudso83

You are one step away from recovering your lost crypto funds. CAPITAL REDEMPTION WIZARD is the solution. Without them, I would have lost everything. I was scammed through a compromised email, lured by a fake Elon Musk Bitcoin investment. After applying and completing KYC, I was credited with 2.5 Bitcoin. When I tried to withdraw, I was hit with fees, and my wallet was emptied of $103,000.00. Their support team was hostile. I researched and found CAPITAL REDEMPTION WIZARD. They recovered my hacked Bitcoin. I learned the Bitcoin offer was a scam. CAPITAL REDEMPTION WIZARD guided me through this. Email: ( [email protected] ) Whats app or text +1 518 468 2985 Website https://bestrecoveryagent.com/
26.09.25 22:25 Slimbella

I fully recommend Sylvester Bryant. He recovered €230,000 in Ethereum for me after I believed it was lost for good. His talent for tracing and reclaiming funds stolen by scammers impresses me deeply. I valued his calm, skilled approach and solid grasp of technology. This gained my confidence at every turn. If scammers have stolen your cash, contact him now. Reach out by email at yt7cracker@gmail. com or via WhatsApp at +1 512 577 7957. He dedicates himself to helping people recover their belongings. His support changed everything for me.
26.09.25 22:59 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.09.25 22:59 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.09.25 03:07 Angela_Moore

I thought my money was lost forever. Then Mr. Sylvester stepped in. He tracked down and recovered $107,000 in stablecoins , plus all my earnings. His talent for finding scam-stolen funds left me stunned. I loved how he handled everything with real expertise and solid tech skills. It got my scammed cash back safe. I trusted him completely. If you've lost funds to a scam, reach out to him now. Email [email protected] or text on WhatsApp at +1 512 577 7957. His support saved me.
28.09.25 00:42 Anitastar

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28.09.25 01:48 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.09.25 01:48 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.09.25 01:49 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13:47 thomassankara

USDT recovery expert / company I thought my money was lost forever. Then Mr. Sylvester stepped in. He tracked down and recovered $107,000 in stablecoins , plus all my earnings. His talent for finding scam-stolen funds left me stunned. I loved how he handled everything with real expertise and solid tech skills. It got my scammed cash back safe. I trusted him completely. If you've lost funds to a scam, reach out to him now. Email [email protected] or text on WhatsApp at +1 512 577 7957. His support saved me.
16:32 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16:32 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19:46 beverlyalex010

In 2022, I lost 1.5 ETH in what I thought was a promising Twitter investment. I reached out to the so-called brokers behind it, but they vanished and stopped replying. For years, I carried the heavy belief that my crypto was gone forever. Then, by chance, I discovered RecoveryFundProvider ( recoveryfundprovider@gmail. com WhatsApp +44 736-644-5035) through a finance podcast. That moment changed everything. Their team didn’t lure me in with false promises or flashy guarantees. Instead, they explained their process clearly specializing in advanced blockchain tracing, smart contract forensics, and even transaction reversal in certain cases. Skeptical but hopeful, I allowed them to review my case. What followed was beyond my expectations. They dug deep into the blockchain ledger, followed the coin’s complex movements across multiple addresses, and even collaborated with global nodes to trace every trail. Two weeks later, I received an alert I never thought I’d see again: my crypto had been fully recovered and safely transferred back to me. I cried tears of relief. What sets RecoveryFundProvider apart isn’t just their cutting-edge expertise, but their unwavering ethics and dedication. They treat every client like more than just another case number even years after a loss. Today, I can confidently say they live up to their name, and i am endlessly grateful.
21:06 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.
21:07 beverlyalex010

I ended my marriage. My spouse was unfaithful. I found clear proof of her cheating. She hid her phone activity. Many secrets were on her phone. She used different apps. I hired a private investigator. I needed solid evidence first. I did not want to accuse her without proof. That could lead to legal trouble. They helped me access her phone. They linked her phone use to mine. I could then watch her phone without her knowing. This is how I learned about her affair. I also found out she planned to take my money. She intended to leave with her new partner. If you are in a similar situation, reach out to [email protected] WhatsApp at +44 736-644-5035. Get ahead of things.